This paper analyzes identifiability and stability for the drifting field underlying distributional matching in the Generative Drifting framework of Deng et al. First, we introduce the class of companion-elliptic kernels, which includes the Laplace kernel and is characterized by a second-order elliptic coupling between each kernel $κ$ in this class and its companion function $η$. For each kernel in this class and each pair of Borel probability measures, we prove that the drifting field vanishes if and only if the two probability measures are equal. We further show that this class consists precisely of Gaussian kernels and Matérn kernels with $ν\ge 1/2$. Second, by constructing counterexamples, we exhibit sequences for which mass escapes to infinity while the field tends to zero; in particular, control of the field norm alone does not guarantee weak convergence. Nevertheless, we prove that the only possible mode of failure is confined to the one-dimensional ray $\{c\,p:0\le c\le 1\}$. Consequently, weak convergence can be restored by imposing an asymptotic lower bound on the intrinsic overlap scalar, a linear observable defined by the kernel and the target measure.

翻译：本文分析了Deng等人提出的生成漂移框架中，实现分布匹配的漂移场的可辨识性与稳定性。首先，我们引入了伴随椭圆核类，该类包含拉普拉斯核，其特点在于该类中的每个核$κ$与其伴随函数$η$之间存在二阶椭圆耦合。对于该类中的每个核及任意一对博雷尔概率测度，我们证明了漂移场为零当且仅当这两个概率测度相等。进一步地，我们证明该类恰好由高斯核和$ν\ge 1/2$的Matérn核构成。其次，通过构造反例，我们展示了质量逃逸至无穷远处而场趋于零的序列；特别是，仅依靠场范数的控制无法确保弱收敛。尽管如此，我们证实可能的失效模式仅限于一维射线$\{c\,p:0\le c\le 1\}$。因此，通过对内蕴重叠标量（由核与目标测度定义的线性可观测量）施加渐近下界，可恢复弱收敛性。